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C is used for an arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus each function has an infinite number of antiderivatives. These formulas only state in another form the assertions in the table of derivatives. Integrals with a singularity[edit] When there is a singularity in the function being integrated such that the integral becomes undefined, i.e., it is not Lebesgue integrable, then C does not need to be the same on both sides of the singularity. The forms below normally assume the Cauchy principal value around a singularity in the value of C but this is not in general necessary. For instance in \int {1 \over x}\,dx = \ln \left|x \right| + C there is a singularity at 0 and the integral becomes infinite there. If the integral above was used to give a definite integral between -1 and 1 the answer would be 0. This however is only the value assuming the Cauchy principal value for the integral around the singularity. If the integration was done in the complex plane the result would depend on the path around the origin, in this case the singularity contributes −iπ when using a path above the origin and iπ for a path below the origin. A function on the real line could use a completely different value of C on either side of the origin as in: \int {1 \over x}\,dx = \ln|x| + \begin{cases} A & \text{if }x>0; \\ B & \text{if }x < 0. \end{cases} Rational functions[edit] More integrals: List of integrals of rational functions These rational functions have a non-integrable singularity at 0 for a ≤ −1. \int k\,dx = kx + C \int x^a\,dx = \frac{x^{a+1}}{a+1} + C \qquad\text{(for } a\neq -1\text{)}\,\! (Cavalieri's quadrature formula) \int (ax + b)^n \, dx= \frac{(ax + b)^{n+1}}{a(n + 1)} + C \qquad\text{(for } n\neq -1\text{)}\,\! \int {1 \over x}\,dx = \ln \left|x \right| + C More generally,[1] \int {1 \over x}\,dx = \begin{cases}\ln \left|x \right| + C^- & x < 0\\ \ln \left|x \right| + C^+ & x > 0 \end{cases} \int\frac{c}{ax + b} \, dx= \frac{c}{a}\ln\left|ax + b\right| + C Exponential functions[edit] More integrals: List of integrals of exponential functions \int e^x\,dx = e^x + C \int f'(x)e^{f(x)}\,dx = e^{f(x)} + C \int a^x\,dx = \frac{a^x}{\ln a} + C Logarithms[edit] More integrals: List of integrals of logarithmic functions \int \ln x\,dx = x \ln x - x + C \int \log_a x\,dx = x\log_a x - \frac{x}{\ln a} + C Trigonometric functions[edit] More integrals: List of integrals of trigonometric functions \int \sin{x}\, dx = -\cos{x} + C \int \cos{x}\, dx = \sin{x} + C \int \tan{x} \, dx = -\ln{\left| \cos {x} \right|} + C = \ln{\left| \sec{x} \right|} + C \int \cot{x} \, dx = \ln{\left| \sin{x} \right|} + C \int \sec{x} \, dx = \ln{\left| \sec{x} + \tan{x}\right|} + C (See Integral of the secant function. This result was a well-known conjecture in the 17th century.) \int \csc{x} \, dx = \ln{\left| \csc{x} - \cot{x}\right|} + C \int \sec^2 x \, dx = \tan x + C \int \csc^2 x \, dx = -\cot x + C \int \sec{x} \, \tan{x} \, dx = \sec{x} + C \int \csc{x} \, \cot{x} \, dx = -\csc{x} + C \int \sin^2 x \, dx = \frac{1}{2}\left(x - \frac{\sin 2x}{2} \right) + C = \frac{1}{2}(x - \sin x\cos x ) + C \int \cos^2 x \, dx = \frac{1}{2}\left(x + \frac{\sin 2x}{2} \right) + C = \frac{1}{2}(x + \sin x\cos x ) + C \int \sec^3 x \, dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan x| + C (see integral of secant cubed) \int \sin^n x \, dx = - \frac{\sin^{n-1} {x} \cos {x}}{n} + \frac{n-1}{n} \int \sin^{n-2}{x} \, dx \int \cos^n x \, dx = \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int \cos^{n-2}{x} \, dx Inverse trigonometric functions[edit] More integrals: List of integrals of inverse trigonometric functions \int \arcsin{x} \, dx = x \arcsin{x} + \sqrt{1 - x^2} + C , \text{ for } \vert x \vert \le +1 \int \arccos{x} \, dx = x \arccos{x} - \sqrt{1 - x^2} + C , \text{ for } \vert x \vert \le +1 \int \arctan{x} \, dx = x \arctan{x} - \frac{1}{2} \ln { \vert 1 + x^2 \vert } + C , \text{ for all real } x \int \arccot{x} \, dx = x \arccot{x} + \frac{1}{2} \ln { \vert 1 + x^2 \vert } + C , \text{ for all real } x \int \arcsec{x} \, dx = x \arcsec{x} - \ln \vert x \, ( 1 + \sqrt{ 1 - x^{-2} } \, ) \vert + C , \text{ for } \vert x \vert \ge +1 \int \arccsc{x} \, dx = x \arccsc{x} + \ln \vert x \, ( 1 + \sqrt{ 1 - x^{-2} } \, ) \vert + C , \text{ for } \vert x \vert \ge +1 Hyperbolic functions[edit] More integrals: List of integrals of hyperbolic functions \int \sinh x \, dx = \cosh x + C \int \cosh x \, dx = \sinh x + C \int \tanh x \, dx = \ln \cosh x + C \int \coth x \, dx = \ln| \sinh x | + C , \text{ for } x \neq 0 \int \operatorname{sech}\,x \, dx = \arctan\,(\sinh x) + C \int \operatorname{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C , \text{ for } x \neq 0 Inverse hyperbolic functions[edit] More integrals: List of integrals of inverse hyperbolic functions \int \operatorname{arsinh} \, x \, dx = x \, \operatorname{arsinh} \, x - \sqrt{ x^2 + 1 } + C , \text{ for all real } x \int \operatorname{arcosh} \, x \, dx = x \, \operatorname{arcosh} \, x - \sqrt{ x^2 - 1 } + C , \text{ for } x \ge 1 \int \operatorname{artanh} \, x \, dx = x \, \operatorname{artanh} \, x + \frac{\ln\left(\,1-x^2\right)}{2} + C , \text{ for } \vert x \vert < 1 \int \operatorname{arcoth} \, x \, dx = x \, \operatorname{arcoth} \, x + \frac{\ln\left(x^2-1\right)}{2} + C , \text{ for } \vert x \vert > 1 \int \operatorname{arsech} \, x \, dx = x \, \operatorname{arsech} \, x + \arcsin x + C , \text{ for } 0 < x \le 1 \int \operatorname{arcsch} \, x \, dx = x \, \operatorname{arcsch} \, x + \vert \operatorname{arsinh} \, x \vert + C , \text{ for } x \neq 0 Products of functions proportional to their second derivatives[edit] \int \cos ax\, e^{bx}\, dx = \frac{e^{bx}}{a^2+b^2}\left( a\sin ax + b\cos ax \right) + C \int \sin ax\, e^{bx}\, dx = \frac{e^{bx}}{a^2+b^2}\left( b\sin ax - a\cos ax \right) + C \int \cos ax\, \cosh bx\, dx = \frac{1}{a^2+b^2}\left( a\sin ax\, \cosh bx+ b\cos ax\, \sinh bx \right) + C \int \sin ax\, \cosh bx\, dx = \frac{1}{a^2+b^2}\left( b\sin ax\, \sinh bx- a\cos ax\, \cosh bx \right) + C Absolute-value functions[edit] Let f be a function which has at most one root on each interval on which it is defined, and g an antiderivative of f that is zero at each root of f (such an antiderivative exists if and only if the condition on f is satisfied), then \int \left| f(x)\right|\,dx = \sgn(f(x))g(x)+C, where sgn(x) is the sign function, which takes the values -1, 0, 1 when x is respectively negative, zero or positive. This gives the following formulas (where a≠0): \int \left| (ax + b)^n \right|\,dx = \sgn(ax + b) {(ax + b)^{n+1} \over a(n+1)} + C \quad [\,n\text{ is odd, and } n \neq -1\,] \,. \int \left| \tan{ax} \right|\,dx = \frac{-1}{a}\sgn(\tan{ax}) \ln(\left|\cos{ax}\right|) + C when ax \in \left( n\pi - \frac{\pi}{2}, n\pi + \frac{\pi}{2} \right) \, for some integer n. \int \left| \csc{ax} \right|\,dx = \frac{-1}{a}\sgn(\csc{ax}) \ln(\left| \csc{ax} + \cot{ax} \right|) + C when ax \in \left( n\pi, n\pi + \pi \right) \, for some integer n. \int \left| \sec{ax} \right|\,dx = \frac{1}{a}\sgn(\sec{ax}) \ln(\left| \sec{ax} + \tan{ax} \right|) + C when ax \in \left( n\pi - \frac{\pi}{2}, n\pi + \frac{\pi}{2} \right) \, for some integer n. \int \left| \cot{ax} \right|\,dx = \frac{1}{a}\sgn(\cot{ax}) \ln(\left|\sin{ax}\right|) + C when ax \in \left( n\pi, n\pi + \pi \right) \, for some integer n. If the function f does not has any continuous anti-derivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then \sgn(f(x)) \int f(x)dx is an anti-derivative of f on every interval on which f is not zero, but may be discontinuous at the points where f(x)=0. For having a continuous anti-derivative, one has thus to add a well chosen step function. If we also use the fact that the absolute values of sine and cosine are periodic with period π, then we get: \int \left| \sin{ax} \right|\,dx = {2 \over a} \left\lfloor \frac{ax}{\pi} \right\rfloor - {1 \over a} \cos{\left( ax - \left\lfloor \frac{ax}{\pi} \right\rfloor \pi \right)} + C\;[citation needed] \int \left|\cos {ax}\right|\,dx = {2 \over a} \left\lfloor \frac{ax}{\pi} + \frac12 \right\rfloor + {1 \over a} \sin{\left( ax - \left\lfloor \frac{ax}{\pi} + \frac12 \right\rfloor \pi \right)} + C\;[citation needed] Special functions[edit] Ci, Si: Trigonometric integrals, Ei: Exponential integral, li: Logarithmic integral function, erf: Error function \int \operatorname{Ci}(x) \, dx = x \operatorname{Ci}(x) - \sin x \int \operatorname{Si}(x) \, dx = x \operatorname{Si}(x) + \cos x \int \operatorname{Ei}(x) \, dx = x \operatorname{Ei}(x) - e^x \int \operatorname{li}(x) \, dx = x \operatorname{li}(x)-\operatorname{Ei}(2 \ln x) \int \frac{\operatorname{li}(x)}{x}\,dx = \ln x\, \operatorname{li}(x) -x \int \operatorname{erf}(x)\, dx = \frac{e^{-x^2}}{\sqrt{\pi }}+x \operatorname{erf}(x) |
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